Mathematical Modeling of the Metastatic Process

Rapid improvements in the detection and tracking of early-stage tumor progression aim to guide decisions regarding cancer treatments as well as predict metastatic recurrence in patients following surgery. Mathematical models may have the potential to further assist in estimating metastatic risk, particularly when paired with in vivo tumor data that faithfully represent all stages of disease progression. Herein we describe mathematical analysis that uses data from mouse models of spontaneous metastasis developing after surgical removal of orthotopically implanted primary tumors. Both presurgical (primary tumor) and postsurgical (metastatic) growth was quantified using bioluminescence and was then used to generate a mathematical formalism based on general laws of the disease (i.e. dissemination and growth). The model was able to fit and predict pre-/post-surgical data at the level of the individual as well as the population. Our approach also enabled retrospective analysis of clinical data describing the probability of metastatic relapse as a function of primary tumor size. In these data-based models, inter-individual variability was quantified by a key parameter of intrinsic metastatic potential. Critically, our analysis identified a highly nonlinear relationship between primary tumor size and postsurgical survival, suggesting possible threshold limits for the utility of tumor size as a predictor of metastatic recurrence. These findings represent a novel use of clinically relevant models to assess the impact of surgery on metastatic potential and may guide optimal timing of treatments in neoadjuvant (presurgical) and adjuvant (postsurgical) settings to maximize patient benefit. Major findings A mathematical model was used to connect presurgical primary tumor volume and postsurgical metastatic burden and survival in two clinically relevant animal models of spontaneous metastasis and one clinical dataset of metastatic relapse probability in breast cancer patients. This model used one specific parameter to quantify differential metastatic aggressiveness, which could be of help for personalizing adjuvant therapy. Simulations revealed a highly nonlinear relationship between resected primary tumor size and metastatic recurrence. These results uncover a computable and patient-dependent threshold for evaluating the efficacy of surgery on overall survival.

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“…More complex considerations on the biology (1,30) and modeling (31) of the metastatic process have been considered elsewhere.…”

Section: Quick Guide To Equations and Assumptionsmentioning

confidence: 99%

Modeling Spontaneous Metastasis following Surgery: An In Vivo-In Silico Approach

et al. 2016

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“…Compared with the numerous models dedicated to tumor growth, relatively few models that describe metastasis have been proposed so far. The existing approaches range from simple phenomenological to complex mechanistic models and contain deterministic and/or stochastic components (14).…”

Section: Major Assumptionmentioning

confidence: 99%

Mathematical Modeling of Tumor Growth and Metastatic Spreading: Validation in Tumor-Bearing Mice

et al. 2014

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Defining tumor stage at diagnosis is a pivotal point for clinical decisions about patient treatment strategies. In this respect, early detection of occult metastasis invisible to current imaging methods would have a major impact on best care and long-term survival. Mathematical models that describe metastatic spreading might estimate the risk of metastasis when no clinical evidence is available. In this study, we adapted a top-down model to make such estimates. The model was constituted by a transport equation describing metastatic growth and endowed with a boundary condition for metastatic emission. Model predictions were compared with experimental results from orthotopic breast tumor xenograft experiments conducted in Nod/Scidg mice. Primary tumor growth, metastatic spread and growth were monitored by 3D bioluminescence tomography. A tailored computational approach allowed the use of Monolix software for mixed-effects modeling with a partial differential equation model. Primary tumor growth was described best by Bertalanffy, West, and Gompertz models, which involve an initial exponential growth phase. All other tested models were rejected. The best metastatic model involved two parameters describing metastatic spreading and growth, respectively. Visual predictive check, analysis of residuals, and a bootstrap study validated the model. Coefficients of determination were R 2 ¼ 0:94 for primary tumor growth and R 2 ¼ 0:57 for metastatic growth. The data-based model development revealed several biologically significant findings. First, information on both growth and spreading can be obtained from measures of total metastatic burden. Second, the postulated link between primary tumor size and emission rate is validated. Finally, fast growing peritoneal metastases can only be described by such a complex partial differential equation model and not by ordinary differential equation models. This work advances efforts to predict metastatic spreading during the earliest stages of cancer. Cancer Res; 74(22); 6397-407. Ó2014 AACR.

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“…Mathematical models of cancer progression have examined avascular growth [16, 17, 3], angiogenesis [18, 19], vascular growth [20, 21], invasion [22, 23] and metastasis [24, 25]; also see these reviews [26, 17, 27, 25]. Tumor metabolism, specifically glycolytic metabolism and its development has also been extensively studied [28, 2].…”

Section: Introductionmentioning

confidence: 99%

Impact of Metabolic Heterogeneity on Tumor Growth, Invasion, and Treatment Outcomes

et al. 2015

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Histopathological knowledge that extensive heterogeneity exists between and within tumors has been confirmed and deepened recently by molecular studies. However, the impact of tumor heterogeneity on prognosis and treatment remains as poorly understood as ever. Using a hybrid multi-scale mathematical model of tumor growth in vascularized tissue, we investigated the selection pressures exerted by spatial and temporal variations in tumor microenvironment and the resulting phenotypic adaptations. A key component of this model is normal and tumor metabolism and its interaction with microenvironmental factors. The metabolic phenotype of tumor cells is plastic, and microenvironmental selection leads to increased tumor glycolysis and decreased pH. Once this phenotype emerges, the tumor dramatically changes its behavior due to acid-mediated invasion, an effect that depends on both variations in the tumor cell phenotypes and their spatial distribution within the tumor. In early stages of growth, tumors are stratified, with the most aggressive cells developing within the interior of the tumor. These cells then grow to the edge of the tumor and invade into the normal tissue using acidosis. Simulations suggest that diffusible cytotoxic treatments such as chemotherapy may increase the metabolic aggressiveness of a tumor due to drug-mediated selection. Chemotherapy removes the metabolic stratification of the tumor and allows more aggressive cells to grow towards blood vessels and normal tissue. Anti-angiogenic therapy also selects for aggressive phenotypes due to degradation of the tumor microenvironment, ultimately resulting in a more invasive tumor. In contrast, pH buffer therapy slows down the development of aggressive tumors, but only if administered when the tumor is still stratified. Overall, findings from this model highlight the risks of cytotoxic and anti-angiogenic treatments in the context of tumor heterogeneity resulting from a selection for more aggressive behaviors.

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Mathematical Modeling of the Metastatic Process

Cited by 20 publications

References 49 publications

Modeling Spontaneous Metastasis following Surgery: An In Vivo-In Silico Approach

Modeling Spontaneous Metastasis following Surgery: An In Vivo-In Silico Approach

Mathematical Modeling of Tumor Growth and Metastatic Spreading: Validation in Tumor-Bearing Mice

Impact of Metabolic Heterogeneity on Tumor Growth, Invasion, and Treatment Outcomes

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